Every Term Meets Every Other Term
Multiplying polynomials looks intimidating until you see the one rule underneath it: every term in the first polynomial multiplies every term in the second, and then you tidy up by combining like terms. That’s the entire method — the distributive property, applied patiently. For the classic two-term case there’s even a nickname for it: FOIL (First, Outer, Inner, Last).
This free polynomial multiplication calculator does the patient part for you. Type both polynomials in plain text — 2x^2+3x+1 and x+4, no special formatting needed — and get the simplified product plus every distribution step shown term by term, so you can follow the method for your own homework.
Use x for the variable and ^ for powers: 2x^2+3x+1. Coefficients can be decimals.
Each polynomial may have degree up to 4 (x^4).
How to Use the Polynomial Multiplication Calculator
- Type the first polynomial. Use x for the variable and ^ for powers: 2x^2+3x+1. Decimal coefficients are fine.
- Type the second polynomial the same way, e.g. x+4. Subtraction is just a negative term: type x-1.
- Click Multiply. The product appears instantly in standard form (highest power first).
- Study the steps. Step 1 shows each term of the first polynomial distributed across the second; Step 2 collects all the partial products; Step 3 combines like terms.
- Check the stats. The degree of the result and the number of partial products are shown — handy for catching mistakes.
What Is Polynomial Multiplication?
A polynomial is a sum of terms like 3x², −x, or 7 — each term is a number (the coefficient) times x raised to a whole-number power (the exponent). Multiplying two polynomials means applying the distributive property: if you have (a + b) × (c + d), you must form all four products ac, ad, bc, bd. With three-term polynomials you get nine products, and so on — the count is always (terms in first) × (terms in second).
After distributing, like terms — terms with the same power of x — get combined by adding their coefficients. Two facts fall out of the method that are worth knowing: the degree of the product is the sum of the degrees (a quadratic times a linear always gives a cubic), and the number of partial products before combining tells you whether you missed a pair.
The Method: Distribute, Then Combine (With a Worked Example)
Exponents add when you multiply: xᵃ × xᵇ = xa+b.
Worked example: (x + 2)(x + 3)
- Distribute the x across (x + 3): x·x = x², x·3 = 3x.
- Distribute the 2 across (x + 3): 2·x = 2x, 2·3 = 6.
- Collect: x² + 3x + 2x + 6 (four partial products — 2 × 2 terms, nothing missed).
- Combine like terms: 3x + 2x = 5x → x² + 5x + 6.
This is exactly what FOIL describes for two binomials: First (x·x), Outer (x·3), Inner (2·x), Last (2·3).
Watch the Signs: A Second Example
Try (2x − 1)(x + 5):
- Distribute 2x: 2x·x = 2x², 2x·5 = 10x.
- Distribute −1: −1·x = −x, −1·5 = −5.
- Collect: 2x² + 10x − x − 5.
- Combine: 10x − x = 9x → 2x² + 9x − 5.
Most errors in polynomial multiplication are sign errors in this step — keeping the minus attached to its term as you distribute is the whole skill.
Special Products: Two Patterns Worth Recognizing
Most products need the full distribute-and-combine routine, but two patterns show up so often that they are worth memorizing as shortcuts — both are just the distributive property pre-combined:
The square of a binomial: (a + b)² = a² + 2ab + b². The result is called a perfect square trinomial. So (x + 5)² expands instantly to x² + 10x + 25, and (2x + 3)² becomes 4x² + 12x + 9. Watch the middle term: it is always twice the product of the two terms, and beginners constantly get its coefficient wrong.
The difference of squares: (a + b)(a − b) = a² − b². The middle terms cancel completely. So (x + 4)(x − 4) = x² − 16. Recognizing this pattern also helps you read factorizations backwards — x² − 49 is (x + 7)(x − 7) on sight.
Limitations
- Single variable, whole-number exponents up to x⁴. The parser understands x only, with exponents 0–4. No y, no x^(1/2), no negative powers — those aren’t polynomials anyway.
- Multiplication only. This tool expands products; it doesn’t divide polynomials or factor them back down.
- Plain-text input has rules. Write powers with ^ (x^2), keep the minus sign glued to its term (x-1, not x – 1 with the sign floating — actually spaces are fine, the parser strips them), and use standard digits. Anything else gets a clear error message instead of a wrong answer.
Tips for Multiplying Polynomials
- Count your partial products. A 3-term polynomial times a 2-term one must produce exactly 6 products before combining. If you have 5, you dropped a pair.
- Check by substituting x = 1. Plug 1 into both originals and into your answer — the products must match. For (x+2)(x+3): 3 × 4 = 12, and x²+5x+6 at x=1 is 1+5+6 = 12. A five-second sanity check.
- Exponents add, coefficients multiply. 3x² × 4x³ = 12x⁵ — never add the coefficients or multiply the exponents.
- Keep minus signs married to their terms. In (2x − 1)(x + 5), the term is −1, not 1. Distribute the sign with the number.
- FOIL is just the distributive property with a nickname. It only covers two binomials — for anything bigger, fall back to “every term times every term” and you’ll never go wrong.
- Write the answer in standard form. Highest exponent first, down to the constant: 2x³ + 11x² + 13x + 4, not the reverse.
More Math Tools
Frequently Asked Questions
What is polynomial multiplication?
It’s the product of two polynomial expressions, found by multiplying every term of the first polynomial by every term of the second (the distributive property) and then combining like terms. For example, (x+2)(x+3) = x²+5x+6.
How does the FOIL method work?
FOIL is a memory aid for multiplying two binomials: multiply the First terms, the Outer terms, the Inner terms, and the Last terms, then combine. For (x+2)(x+3): x·x + x·3 + 2·x + 2·3 = x²+5x+6.
Can this calculator multiply three polynomials at once?
Not directly — it multiplies two at a time. To handle three, multiply the first two, then multiply that result by the third using the calculator again.
Why does the degree of the product equal the sum of the degrees?
Because the highest-power term of the product comes from multiplying the highest-power terms of the inputs, and exponents add: x² × x³ = x⁵. So a degree-2 polynomial times a degree-3 one always gives degree 5.
How do I type exponents and negative terms?
Use the caret for powers (2x^2+3x+1) and type subtraction normally (x-1). The parser also accepts superscript characters like x² if you paste them in.
How can I check my answer?
Substitute x = 1 into both original polynomials and into your answer — the two products must agree. You can also substitute x = 0, which should give (constant₁ × constant₂) as the answer’s constant term.
What is the difference between expanding and factoring?
Expanding (what this calculator does) turns a product like (x+2)(x+3) into a sum, x²+5x+6. Factoring is the reverse journey: turning x²+5x+6 back into (x+2)(x+3).
What is the box method for multiplying polynomials?
A layout trick: write one polynomial’s terms along the top of a grid and the other’s down the side, fill each cell with the term-by-term product, then add the diagonals to combine like terms. It is the same distributive mathematics as the standard method — it just organizes the partial products so none get lost.
What is a perfect square trinomial?
The product of a binomial multiplied by itself: (a + b)² = a² + 2ab + b². For example, (x + 5)² = x² + 10x + 25. Once you recognize the pattern you can expand it instantly — just be careful with the middle term, which is always twice the product of the two terms.